Beyond monuments, we know a very great deal about Roman history from written histories. The major sources (at least, in my estimation) are:

Livy, who wrote a history ranging from the founding of Rome to the time of Augustus.

Tacitus, who wrote several histories but most notably the Annals, which covers the reigns of Tiberius through to Nero.

Suetonius, who wrote biographies of all Emperors through to Domitian.

Cassius Dio, who also write biographies of many Emperors.

There are several considerations, however, in the use of these sources. Much of Livy's history, which pertains to Rome's legendary early years, has to be taken with a pinch of salt since much of it is impossible to verify and isn't really credible. We should also be aware of potential biases/agendas of these historians. For example, we might consider that Tacitus's account of Nero's reign is coloured by his moral disapproval of Nero's character.

There are many other, lesser histories as well, of greater and lesser reliability/usefulness. Other written sources that aren't directly histories, such as Cicero's speeches, can also provide historical insight.

Besides monuments, there are also many other forms of archaeological evidence. One of the most notable is coins. Every Emperor had coins minted in their honour, and while this isn't really my field my understanding is that they used coins to boast about their achievements. Coins also serve to verify the dates of the reigns of Emperors.

I second that answer; collected letters from those citizens who actually experienced the times (Atticus, Cicero) as well as biographies by their contemporaries or near-contemporaries (such as Nepos) are the best sources; historiography has to be taken with a grain of salt, considering not only the biases of the original writer, but also any translation biases (if dealing with an English translation). Nepos, Dio, Suetonius, Tacitus, and Livy are good for the time up until Augustus; after the advent of Christ, I'd tend more toward Christian scholars.

There's certainly not a dearth, so to speak, of primary resources regarding Rome in pre-Augustan and Augustan times; traditions vary when speaking of the founding of Rome itself, as well as the period when Rome was a monarchy (in fairly loose terms), but once you hit the Caesars, historiography tends to flourish, however accurate it may or may not be.

You can also learn a lot from various other art (sculpture, mosaics, wall paintings, small scale terracotta figures etc). These sources have in various ways helped contribute a lot of information to the historical record, especially about the daily lives of the average citizens, and are good to use together with the literary sources. Grave inscriptions can also provide a great deal of information about the Roman population at different times, often recording info about age of death, occupation, status (slave or freedman), marital status, number of children, and sometimes even cause of death, which again is useful stuff to know if you are trying to build a clearer picture of what Rome was like.